#27290311

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10917

Subgroups

	subgroup	G/H
center	C₄	#16810499
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#344	C₂
socle	C₂×C₅₃	#1436
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#344	D₅₃
hypocenter	C₅₃	#10917
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#344

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(54,57,58,69,59,68,70,81)(55,61,62,75,63,74,76,84)(56,65,67,80,66,79,71,82)(60,72,73,83,64,77,78,85);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,56)(55,60,63,64)(57,65)(58,67)(59,66)(61,72,74,77)(62,73,76,78)(68,79)(69,80)(70,71)(75,83,84,85)(81,82);(54,55,58,62,59,63,70,76)(56,64,67,78,66,60,71,73)(57,61,69,75,68,74,81,84)(65,72,80,83,79,77,82,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=g,c^2=f,f^2=h,b^a=bd,c^a=ce,g^a=gh,d^b=dh,e^b=eh,i^b=i^52,d^c=dh,i^c=i^52

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 75787);